Structure learning is the task of finding the dependency graph of a Markov random field (MRF) given i.i.d.

Probabilistic graphical models have been widely used to model complex systems and aid scientific discoveries. As a result, there is a large body of literature focused on consistent model selection. However, scientists are often interested in understanding uncertainty associated with the estimated parameters, which current literature has not addressed thoroughly. In this paper, we propose a novel estimator for edge parameters for pairwise graphical models based on Hyv\arinen scoring rule. Hyv\arinen scoring rule is especially useful in cases where the normalizing constant cannot be obtained efficiently in a closed form.

Neural Information Processing Systems http://nips.cc/

I think the proposed graphical model is interesting and novel. The authors provide a lower bound and a fully convex upper bound for partition function. In particular, variational upper/lower bound are formulated by using pairwise graphical models, e.g. The author also provides algorithms, Frank-Wolfe, PGD and BP, for finding upper/lower bound. The experimental results shows that cooperative graphical models and its variational methods performs better than pairwise graphical models for image segmentation.

This paper studies the problem of estimating parameters in a pairwise graphical model and constructing confidence intervals for the parameters. As part of this an asymptotically normal estimator is constructed. The key progress made in this paper is that inference (i.e., confidence intervals) is done in a setting where computation of basic quantities (e.g. Specifically, an estimator based on Hyvarinen score is given for estimation of a single edge in the pairwise undirected graphical model. The new scoring rule uses the conditional density of two variables given the rest. A first step forms a preliminary Markov blanket for a pair of variables, and the estimator then re-optimizes over parameter value, which has a sort of decoupling effect.

Probabilistic graphical models are a powerful concept for modeling high-dimensional distributions. Besides modeling distributions, probabilistic graphical models also provide an elegant framework for performing statistical inference; because of the high-dimensional nature, however, one must often use approximate methods for this purpose. Belief propagation performs approximate inference, is efficient, and looks back on a long success-story. Yet, in most cases, belief propagation lacks any performance and convergence guarantees. Many realistic problems are presented by graphical models with loops, however, in which case belief propagation is neither guaranteed to provide accurate estimates nor that it converges at all. This thesis investigates how the model parameters influence the performance of belief propagation. We are particularly interested in their influence on (i) the number of fixed points, (ii) the convergence properties, and (iii) the approximation quality.

We consider the problem of learning Markov Random Fields (including the prototypical example, the Ising model) under the constraint of differential privacy. Our learning goals include both structure learning, where we try to estimate the underlying graph structure of the model, as well as the harder goal of parameter learning, in which we additionally estimate the parameter on each edge. We provide algorithms and lower bounds for both problems under a variety of privacy constraints -- namely pure, concentrated, and approximate differential privacy. While non-privately, both learning goals enjoy roughly the same complexity, we show that this is not the case under differential privacy. In particular, only structure learning under approximate differential privacy maintains the non-private logarithmic dependence on the dimensionality of the data, while a change in either the learning goal or the privacy notion would necessitate a polynomial dependence. As a result, we show that the privacy constraint imposes a strong separation between these two learning problems in the high-dimensional data regime.

Probabilistic graphical models have been widely used to model complex systems and aid scientific discoveries. As a result, there is a large body of literature focused on consistent model selection. However, scientists are often interested in understanding uncertainty associated with the estimated parameters, which current literature has not addressed thoroughly. In this paper, we propose a novel estimator for edge parameters for pairwise graphical models based on Hyv\"arinen scoring rule. Hyv\"arinen scoring rule is especially useful in cases where the normalizing constant cannot be obtained efficiently in a closed form.

We characterize the effectiveness of a classical algorithm for recovering the Markov graph of a general discrete pairwise graphical model from i.i.d. samples. The algorithm is (appropriately regularized) maximum conditional log-likelihood, which involves solving a convex program for each node; for Ising models this is $\ell_1$-constrained logistic regression, while for more general alphabets an $\ell_{2,1}$ group-norm constraint needs to be used. We show that this algorithm can recover any arbitrary discrete pairwise graphical model, and also characterize its sample complexity as a function of model width, alphabet size, edge parameter accuracy, and the number of variables. We show that along every one of these axes, it matches or improves on all existing results and algorithms for this problem. Our analysis applies a sharp generalization error bound for logistic regression when the weight vector has an $\ell_1$ constraint (or $\ell_{2,1}$ constraint) and the sample vector has an $\ell_{\infty}$ constraint (or $\ell_{2, \infty}$ constraint). We also show that the proposed convex programs can be efficiently solved in $\tilde{O}(n^2)$ running time (where $n$ is the number of variables) under the same statistical guarantees. We provide experimental results to support our analysis.

We propose a new family of message passing techniques for MAP estimation in graphical models which we call {\em Sequential Reweighted Message Passing} (SRMP). Special cases include well-known techniques such as {\em Min-Sum Diffusion} (MSD) and a faster {\em Sequential Tree-Reweighted Message Passing} (TRW-S). Importantly, our derivation is simpler than the original derivation of TRW-S, and does not involve a decomposition into trees. This allows easy generalizations. We present such a generalization for the case of higher-order graphical models, and test it on several real-world problems with promising results.